Knowing the integral property: $$\int_{-\infty}^tx(\tau)\,d\tau\,\overset{F}\longleftrightarrow\,\frac{X(\omega)}{j\omega}\,+\,\pi X(0)\delta(\omega)$$ and the duality property: $$f(t)\,\overset{F}\longleftrightarrow\,g(\omega)\,\Rightarrow\,g(t)\,\overset{F}\longleftrightarrow\,2\pi f(-\omega)$$

How can we prove that: $$\frac{-x(t)}{jt}\,+\,\pi x(0) \delta(t)\,\overset{F}\longleftrightarrow\,\int_{-\infty}^\omega X(\eta)\,d\eta$$

Here the $\,\overset{F}\longleftrightarrow\,$ denotes the Fourier transform defined as: $$X(\omega)\,=\,\int_{-\infty}^\infty\,x(t)\,e^{-j\omega t}\,dt$$So:$$\Rightarrow\,x(t)\,=\,\frac{1}{2\pi}\,\int_{-\infty}^\infty\,X(\omega)\,e^{j\omega t}\,d\omega$$ My question stems from the signals and systems course in the electrical engineering field, so here the Fourier transform should give us all of the available frequencies in the signal $x(t)$ ($x(t)$ models the signal's behavior in the time dimension): $$X(\omega)=2\pi.(\frac{available\,frequency}{d\omega})=2\pi.(frequency\,density)$$ Also the $\delta(x)$ is the Dirac delta function.
Here are two other properties that might be helpful in solving this question: $$x(-t)\,\overset{F}\longleftrightarrow\,X(-\omega)$$ $$x^*(t)\,\overset{F}\longleftrightarrow\,X^*(-\omega)$$ By $x^*(t)$ I mean the complex conjugate of $x(t)$.
Let me know if anything needs further explanations.

Knowing the integral property: $$\int_{-\infty}^tx(\tau)\,d\tau\,\overset{F}\longleftrightarrow\,\frac{X(\omega)}{j\omega}\,+\,\pi X(0)\delta(\omega)$$ and the inversion theorem: $$f(t)\,\overset{F}\longleftrightarrow\,g(\omega)\,\Rightarrow\,g(t)\,\overset{F}\longleftrightarrow\,2\pi f(-\omega)$$

How can we prove that: $$\frac{-x(t)}{jt}\,+\,\pi x(0) \delta(t)\,\overset{F}\longleftrightarrow\,\int_{-\infty}^\omega X(\eta)\,d\eta$$

Here the $\,\overset{F}\longleftrightarrow\,$ denotes the Fourier transform defined as: $$X(\omega)\,=\,\int_{-\infty}^\infty\,x(t)\,e^{-j\omega t}\,dt$$So:$$\Rightarrow\,x(t)\,=\,\frac{1}{2\pi}\,\int_{-\infty}^\infty\,X(\omega)\,e^{j\omega t}\,d\omega$$ My question stems from the signals and systems course in the electrical engineering field, so here the Fourier transform should give us all of the available frequencies in the signal $x(t)$ ($x(t)$ models the signal's behavior in the time dimension): $$X(\omega)=2\pi.(\frac{available\,frequency}{d\omega})=2\pi.(frequency\,density)$$ Also the $\delta(x)$ is the Dirac delta function.
Here are two other properties that might be helpful in solving this question: $$x(-t)\,\overset{F}\longleftrightarrow\,X(-\omega)$$ $$x^*(t)\,\overset{F}\longleftrightarrow\,X^*(-\omega)$$ By $x^*(t)$ I mean the complex conjugate of $x(t)$.
Let me know if anything needs further explanations.

Knowing the integral property: $$\int_{-\infty}^tx(\tau)\,d\tau\,\overset{F}\longleftrightarrow\,\frac{X(\omega)}{j\omega}\,+\,\pi X(0)\delta(\omega)$$ and the duality property: $$f(t)\,\overset{F}\longleftrightarrow\,g(\omega)\,\Rightarrow\,g(t)\,\overset{F}\longleftrightarrow\,2\pi f(-\omega)$$

How can we prove that: $$\frac{-x(t)}{jt}\,+\,\pi x(0) \delta(t)\,\overset{F}\longleftrightarrow\,\int_{-\infty}^\omega X(\eta)\,d\eta$$

Knowing the integral property: $$\int_{-\infty}^tx(\tau)\,d\tau\,\overset{F}\longleftrightarrow\,\frac{X(\omega)}{j\omega}\,+\,\pi X(0)\delta(\omega)$$ and the duality property: $$f(t)\,\overset{F}\longleftrightarrow\,g(\omega)\,\Rightarrow\,g(t)\,\overset{F}\longleftrightarrow\,2\pi f(-\omega)$$

How can we prove that: $$\frac{-x(t)}{jt}\,+\,\pi x(0) \delta(t)\,\overset{F}\longleftrightarrow\,\int_{-\infty}^\omega X(\eta)\,d\eta$$

Here the $\,\overset{F}\longleftrightarrow\,$ denotes the Fourier transform defined as: $$X(\omega)\,=\,\int_{-\infty}^\infty\,x(t)\,e^{-j\omega t}\,dt$$So:$$\Rightarrow\,x(t)\,=\,\frac{1}{2\pi}\,\int_{-\infty}^\infty\,X(\omega)\,e^{j\omega t}\,d\omega$$ My question stems from the signals and systems course in the electrical engineering field, so here the Fourier transform should give us all of the available frequencies in the signal $x(t)$ ($x(t)$ models the signal's behavior in the time dimension): $$X(\omega)=2\pi.(\frac{available\,frequency}{d\omega})=2\pi.(frequency\,density)$$ Also the $\delta(x)$ is the Dirac delta function.
Here are two other properties that might be helpful in solving this question: $$x(-t)\,\overset{F}\longleftrightarrow\,X(-\omega)$$ $$x^*(t)\,\overset{F}\longleftrightarrow\,X^*(-\omega)$$ By $x^*(t)$ I mean the complex conjugate of $x(t)$.
Let me know if anything needs further explanations.